We extend the probability-time framework—defined as the information-geometric arc-length of irreversible trajectories in state space— to a two-qubit open-system setting and sharpen its interpretation using a cross-metric amplitude-damping simulation. Probability-time is defined as the accumulated information-geometric arc-length of the irreversible component of a trajectory in density-operator space. For a Bell-state initial condition evolved under local dephasing and local amplitude damping, the dephasing case shows approximate collapse of concurrence curves when expressed against probability-time, whereas amplitude damping does not. The latest simulation further shows that, for the same two-qubit amplitude-damping trajectory, both the Bures--Helstrom and Kubo--Mori constructions produce monotone probability-time while yielding different numerical parameterizations. This supports a more precise claim: the metric-independent content of probability-time is the induced irreversible ordering, while the numerical parameterization remains metric-dependent.
We extend the probability-time framework—defined as the information-geometric arc-length of irreversible trajectories in state space— to a two-qubit open-system setting and sharpen its interpretation using a cross-metric amplitude-damping simulation. Probability-time is defined as the accumulated information-geometric arc-length of the irreversible component of a trajectory in density-operator space. For a Bell-state initial condition evolved under local dephasing and local amplitude damping, the dephasing case shows approximate collapse of concurrence curves when expressed against probability-time, whereas amplitude damping does not. The latest simulation further shows that, for the same two-qubit amplitude-damping trajectory, both the Bures--Helstrom and Kubo--Mori constructions produce monotone probability-time while yielding different numerical parameterizations. This supports a more precise claim: the metric-independent content of probability-time is the induced irreversible ordering, while the numerical parameterization remains metric-dependent.
This paper develops a mathematical foundation for probability-time, an ordering parameter that measures irreversible progression along quantum state-space trajectories distinct from chronological time. In the setting of finite-dimensional CPTP dynamics with nonnegative entropy production, probability-time is defined as the accumulated information-geometric arc length of the entropy-producing component of the system’s progression. The paper describes an axiomatic characterization that isolates the invariant content of the construction: monotonicity on dissipative trajectories, stationarity under reversible evolution, and equivalence under strictly monotone reparameterizations. Within this framework, admissible contractive monotone metrics are shown to generate a common irreversible ordering while producing generally distinct numerical parameterizations. For commuting trajectories, the metrics yield only metric-dependent scalings, whereas the noncommuting trajectories exhibit genuinely nonlinear metric dependence. In detailed-balance regimes the relation between probability-time and Spohn entropy production is analyzed, demonstrating the failure of global strong canonicality but establishing a near-equilibrium quadratic relation under the Kubo–Mori metric. We thereby clarify in what sense probability-time provides a geometric measure of irreversible progression.
We develop a Fisher-informational reformulation of physical time in which clock time is not regarded as a fundamental ontological substance, but as an emergent calibration of causally ordered distinguishability among physical states. The operational starting point is that clocks do not measure time itself; rather, they instantiate reproducible physical processes whose distinguishable states are correlated with other events. We introduce a causal-informational parameter, denoted by Lambda_F, defined as an accumulated Fisher-geometric distance along a causally admissible trajectory in state space. In classical statistical systems, this parameter is generated by the Fisher information metric; in quantum systems, the corresponding construction is associated with quantum Fisher information, the Bures metric, and the Fubini-Study geometry of projective Hilbert space. The manuscript distinguishes model-dependent Fisher information from quantum Fisher information, clarifies the reparameterization of Schrodinger dynamics, and gives explicit examples involving a qubit clock, an exponential decay process, and a Fisher characterization of clock quality. The proposal is positioned relative to relational time, the Page-Wootters mechanism, thermal time, quantum speed-limit relations, information geometry, and the problem of time in quantum gravity. We do not claim that relational or emergent time is new. The specific contribution is the use of Fisher distinguishability as an operational precursor from which ordinary clock time can be reconstructed. In this sense, the central statement of the paper is: time is not measured by clocks; clock time is reconstructed from the Fisher distinguishability accumulated along causally ordered physical changes.
We develop an information-geometric framework for analyzing quantum state-update processes generated by explicit collapse dynamics. Whereas a purely geometric description characterizes collapse as a finite displacement between an initial and a final statistical state, the present approach incorporates the dynamical law governing the intermediate evolution. A quantum state is represented by a density operator ρ(t) satisfying a specified dynamical equation, dρdt=C[ρ,t], or, more generally, by a unitary-plus-collapse evolution in which the collapse contribution is explicitly identified. The resulting trajectory is embedded in the statistical manifold of quantum states and analyzed using the quantum Fisher information metric. This allows one to define the instantaneous Fisher speed, the information-geometric length of the collapse trajectory, and an integrated geometric cost associated with the dynamical state change. The framework also permits comparison between the actual trajectory generated by a collapse model and the corresponding Fisher-geodesic trajectory connecting the same initial and final states. Relative entropy provides a complementary global measure of the distinguishability generated by the evolution, while the quantum Fisher metric characterizes its local statistical structure. The formalism is constructed so that different dynamical collapse proposals can be analyzed within a common geometric language without assuming that the geometric quantities themselves constitute a collapse mechanism: this is a diagnostic framework rather than a mechanism. In particular, once a specific collapse generator is supplied, the framework can determine how its parameters, localization properties, stochasticity, and time scale are reflected in the geometry of the associated state trajectory. The resulting formulation therefore provides a bridge between dynamical models of wave-function collapse and the information geometry of a quantum state space, while preserving the distinction between a physical collapse law and its geometric characterization.
Can spatial distance and proper time be defined operationally — by how a system leaks information to its environment — rather than postulated as primitive geometric quantities? This paper answers affirmatively for both, within Markovian open quantum dynamics, building on the survival-expansion framework of the companion paper "Local Curvature of Quantum Information" (doi:10.5281/zenodo.18792424). Spatial distance is defined as the excess irreversibility generated by superposing two states: when a system is superposed over two locations, the environment learns which at a rate growing with their separation. This distance is proven to be a genuine metric — it satisfies the triangle inequality via an isometric embedding into environmental "shadow space", recovers the Euclidean metric of R³ for independent position channels, and becomes undefined before decoherence even for orthogonal states, so that geometry and classicality emerge together. Proper time is defined as elapsed background time minus the minimal dissipation needed to traverse a worldline. The reversed triangle inequality of relativity (the twin paradox) then follows automatically from the subadditivity of optimal costs, and the exact Minkowski interval corresponds to a dissipation rate that saturates at high transport velocity: the light cone as a bound on irreversibility production. Both Lorentzian signatures are traced to a single convex-analytic root: spatial distance is directly a subadditive object (a norm), while proper time is an additive duration minus a subadditive optimal cost, hence superadditive. In an exactly solvable driven damped-oscillator model, the information deficit coincides identically with Clausius entropy production, the minimal transport deficit obeys Λ∗=2kBTbγmTΔx2, and the emergent light speed is c2=γmκkBTb=κD, with D the Einstein diffusion coefficient. The universality of the emergent light cone is shown to be equivalent to a Compton-clock scaling of the dissipation–time conversion rate (κ=ℏ2mc2), with the quantum–classical crossover condition kBTb=2ℏγ arising as an internal consistency requirement rather than a postulate. A junction-cost theorem establishes that velocity discontinuities incur strictly positive excess dissipation, so the turning twin ages strictly less even at equal speeds. All results are verified numerically on lattice decoherence models, exact linear–quadratic optimal control, and Fock-space computations; the verification scripts accompany the manuscript. Locality of the environmental couplings is assumed, not derived, and the construction has analog-gravity status: a Lorentzian interval in a preferred frame defined by the bath.
Abstract Understanding thermalization and irreversibility in open quantum systems requires char-acterizing the competition between coherent unitary dynamics and dissipative environmen-tal interactions. In this work, we present a systematic numerical study of thermalization dynamics in a two-qubit open quantum system governed by a completely positive and trace-preserving Lindblad master equation. The coherent evolution rate is controlled by a real scaling parameter α multiplying the Hamiltonian commutator, while decoherence is indepen-dently controlled by a dephasing rate γ . Although such a scaling of coherent dynamics can be absorbed into a rescaling of time in isolated systems, it becomes physically meaningful in open systems where decoherence introduces an absolute timescale. We investigate how the relative strength of coherent evolution and decoherence influences observable dynamics, purity decay, entropy production, and thermalization times. Using robust local and two-site observables, we show that varying α at fixed γ produces distinct transient trajectories, even though all evolutions converge to the same steady state. We further define an operational thermalization time τ ( γ, α ) based on a purity threshold and map the resulting thermalization-time surface across a broad parameter range. Increasing α generally accelerates oscillatory dynamics and modifies the approach to equilibrium, leading to measurable shifts in thermalization times without altering asymptotic states. These findings provide a clear and physically consistent characterization of how Hamil-tonian strength and decoherence rate jointly determine thermalization behavior in open quantum systems. The framework offers a practical parameterization for numerical mod-eling and experimental studies of coherent–dissipative competition in quantum information processing, quantum control, and open-system dynamics.